Let
B
=
B
t
1
,
…
,
B
t
d
t
≥
0
be a
d
-dimensional bifractional Brownian motion and
R
t
=
B
t
1
2
+
⋯
+
B
t
d
2
be the bifractional Bessel process with the index
2
HK
≥
1
. The Itô formula for the bifractional Brownian motion leads to the equation
R
t
=
∑
i
=
1
d
∫
0
t
B
s
i
/
R
s
d
B
s
i
+
HK
d
−
1
∫
0
t
s
2
HK
−
1
/
R
s
d
s
. In the Brownian motion case
K
=
1
and
H
=
1
/
2
,
X
t
≔
∑
i
=
1
d
∫
0
t
B
s
i
/
R
s
d
B
s
i
,
d
≥
1
is a Brownian motion by Lévy’s characterization theorem. In this paper, we prove that process
X
t
is not a bifractional Brownian motion unless
K
=
1
and
H
=
1
/
2
. We also study some other properties and their application of this stochastic process.